Allow X to signify the voltage at the yield of a mouthpiece, and assume that X has a uniform circulation on the stretch from ?1 to 1. The voltage is handled by a "hard limiter" with cutoff esteems ?0.5 and 0.5, so the limiter yield is an arbitrary variable Y identified with X by Y = X if |X| ? 0.5, Y = 0.5 if X > 0.5, and Y = ?0.5 if X

(a) What is P(Y = 0.5)?

(b) Obtain the combined appropriation capacity of Y.

If it's not too much trouble, show your work, much obliged!

A firm uses pattern projection and occasional components to reproduce deals for a given time frame period. It doles out "0" if deals fall, "1" if deals are consistent, "2" if deals rise reasonably, and "3" if deals rise a ton. The test system creates the accompanying yield.

0 1 0 2 0 1 0 3 2 0 2 1 2 3 1 2 0 2 0 3 0 2 1 0 1

Gauge the likelihood that deals will rise tolerably.

a

0.258

b

0.233

c

0.312

@2@

d

0.226

Consider the examination of flipping a reasonable coin until two heads or two tails show up in

progression.

(a) Describe the example space.

(b) What is the likelihood that the analysis closes before the 6th throw?

(c) What is the likelihood that the investigation closes after a much number of throws?

(d) Given that the examination closes with two heads, what is the likelihood that the

analyze closes before the 6th throw?

(e) Given that the investigation doesn't end before the third throw, what is the likelihood that

the analysis doesn't end after the 6th throw?

Allow X to mean the measure of time a book on two-hour hold is really looked at, and assume the cdf is the accompanying. F(x) = 0 if x = 5. Utilize the cdf to acquire the accompanying. (On the off chance that vital, round your response to four decimal spots.)

(g) Calculate V(X) and ?x.

(h) If the borrower is charged a sum h(X) = X2 when checkout length is X, figure the normal charge E[h(X)].

Problem 2: (a) Let X and Y be jointly Gaussian random variables with means Ax and by and standard deviations ox and dy respectively, and correlation coefficient pry. Define new random variables U, and Uz by the transformation. U1 = (X - px)/ox + (Y - ur)/oy U2 = (X - Hx)/Ox - (Y -My)/oy Show that U, and Uz are Gaussian random variables with zero means and zero correlation coefficient. How would you modify the transformation to obtain two new independent Gaussian random variables Vi and Va with zero means and unit variances? Note: The above illustrates how you can transform two correlated Gaussian random variables X and Y to two uncorrelated and independent standard normal Gaussian random variables V, and Va. (b) Given two independent standard normal Gaussian random variables Z1 and Zz, show that you can transform them to two correlated Gaussian random varibles X - Gaussian(ux, Gx) and Y ~ Gaussian(ay, oy) with correlation coefficient pxy by the transformation. X = ZOx+ Hx Y = PxYoy ZI + \\/1 - pxroy Zz + ur Calculate EX], E[Y], Var(X), Var(Y) and E[XY] using the formulas above.The probability density function (pdf) of a Gaussian random variable is: P(I) = C (1) V2no2 where a is the mean of the random variable, and o is the standard deviation. (1) Please plot the pdf of a Gaussian random variable (the height of a American guy) in Matlab, if we know the mean is 5 feet 9 inches, and the standard deviation is 1 foot. (2) Please generate a large number of instances of such a Gaussian random variable in Matlab using the randn command, then estimate the pdf from the instances you gen- crated (hint: use hist command to get histogram then scale it). (3) Please plot the theoretical pdf and the estimated pdf in one plot (do they match?). In your report, please include the three plots and the Matlab program (in the ap- pendix).6. For a random variable x with PDF : 1 Px (x) = = e 32 u(x) 2V2x a. Sketch px (x) and state with reasons if this is a Gaussian random variable. ( 5 points). b. Determine (1) P(x 2 1) and (1) P(1